Reverse Mortgage Pricing and Risk Analysis Allowing for Idiosyncratic House Price Risk and Longevity Risk

1/17 Reverse Mortgage Pricing and Risk Analysis Allowing for Idiosyncratic House Price Risk and Longevity Risk Adam Wenqiang Shao, Katja Hanewald, Michael Sherris ARC Centre of Excellence in Population Ageing Research (CEPAR) and School of Risk & Actuarial Studies, University of New South Wales, Sydney 9 th International Longevity Risk and Capital Markets Solutions Conference 6-7 Sep, 2013, Beijing

2/17 Topic Coverage 1 Introduction 2 Pricing Framework 3 Idiosyncratic House Price Risk 4 Termination Model 5 Pricing and Risk Analysis 6 Conclusions

3/17 Topic coverage 1 Introduction 2 Pricing Framework 3 Idiosyncratic House Price Risk 4 Termination Model 5 Pricing and Risk Analysis 6 Conclusions

3/17 Research Motivation Population ageing: how to finance health care in retirement? large component of household wealth in home equity role for reverse mortgages and other equity release products Growing literature on reverse mortgage pricing and risk management impact of longevity risk (Wang et al., 2008; Li et al., 2010; Yang, 2011) new pricing framework (Alai et al., 2013; Cho et al., 2013) house price risk typically analysed using time series models based on one market-wide index (e.g., Chen et al., 2010; Yang, 2011; Lee et al., 2012) This study: idiosyncratic house price risk and longevity risk portfolio of options (not: option on portfolio) idiosyncratic house price risk substantial (Hanewald and Sherris, 2013) disaggregated house price indices: more volatility and different trends (Shao et al., 2013)

4/17 Topic coverage 1 Introduction 2 Pricing Framework 3 Idiosyncratic House Price Risk 4 Termination Model 5 Pricing and Risk Analysis 6 Conclusions

Accumulated loan amount Consider a reverse mortgage loan with: variable interest rate lump-sum payment single borrower The accumulated loan amount at termination: L Tx = L 0 exp ( Kx +1 t=1 ( y 1 t + ϕ + π )) 4/17 T x : random termination time for age (x), K x = [T x ] L 0 : initial loan amount yt 1 : quarterly risk-free rate ϕ: quarterly lending margin π: mortgage insurance premium

No-Negative Equity Guarantee (NNEG) At termination, the repayment amount is min{l Tx, H Tx } Borrower s net equity is: Present value of lender s loss: Net Equity Tx = H Tx min{l Tx, H Tx } = max{h Tx L Tx, 0} K x +1 Loss Tx = max{l Tx (1 c)h Tx, 0} c: transaction cost m s : risk-adjusted stochastic discount factor s=1 m s The value of NNEG is: NNEG = ω x 1 t=0 E [ t q c x Loss t+1 ] t q c x : probability of terminating between t and t + 1 (1)

6/17 Mortgage insurance premium Premium accumulation: dp t = πl t dt EPV of charged premium is: w x 1 MIP = π E t=0 [ tp c x L t ] t m s s=0 (2) tp c x : in-force probability Solve for π by equating expectations of MIP and NNEG: Equations (1) and (2). SF = ω x 1 t=0 { t q c x E [Loss t+1 ] t p c x πe [ L t s=0 ]} t m s (3)

7/17 Topic coverage 1 Introduction 2 Pricing Framework 3 Idiosyncratic House Price Risk 4 Termination Model 5 Pricing and Risk Analysis 6 Conclusions

7/17 Hybrid house price model (Shao et al., 2013) V it = α + T β + X γ + X T + η i + ξ it (4) β: coefficients for time dummy variables T γ: coefficients for house characteristic variables X : coefficients for the interactions between time dummy and house characteristic variables η i : individual house specific error, uncorrelated with ξ Differencing Equation (4): V jt V js = D β + X D + ξ jt ξ js (5) D: differenced time dummy variables No data on changes of X

8/17 Economic scenario generation Model: VAR(2) State variables included in Y t : Y t = κ + Φ 1 Y t 1 + Φ 2 Y t 2 + Σ 1/2 Z t 1 aggregate house price index growth rates 2 rental yields 3 GDP growth rates 4 1-quarter zero-coupon bond yield rates 5 spread of 5-year over 1-quarter bond yield rates Z t : vector of independent standard normal variables Derive bond yield curve based on the development of the state variables under the VAR(2) model. Derive risk-adjusted stochastic discount factors (m t ) assuming no arbitrage (Cochrane and Piazzesi, 2002; Alai et al., 2013).

Projection of individual house prices Model: VARX(1,0) IHG t = κ + ΦIHG t 1 + βhg t + Σ 1/2 Z t, IHG t : disaggregated house price growth rate HG t : aggregate house price growth rate Aggregate Central Business District 3000 Historical Index Forecasted Index 3000 Historical Index Forecasted Index 2000 2000 1000 1000 /17 0 1990 2000 2010 2020 Time 0 1990 2000 2010 2020 Time

10/17 Topic coverage 1 Introduction 2 Pricing Framework 3 Idiosyncratic House Price Risk 4 Termination Model 5 Pricing and Risk Analysis 6 Conclusions

Termination triggers (Ji et al., 2012) Four major triggers of repayment: λ: age-specific factor to scale down at-home mortality κ: age-specific factor that relates LTC incidence to mortality q pre i : duration-dependent annual prepayment probability : duration-dependent annual refinancing probability q ref i In-force probability of the reverse mortgage contract is: { t } t tpx c = exp (λ x+s + κ x+s )ˆµ x+s ds 0 i=1 [ (1 q pre i )(1 q ref i )] 1/4 (6) 0/17

Stochastic mortality: two factor CBD model logit q(t, x) = κ (1) t + κ (2) t (x x) (7) q(t, x): death probability x: average age 100 100 90 90 Age 80 70 Age 80 70 60 60 50 1970 1980 1990 2000 Time 50 1970 1980 1990 2000 Time Males (left) and females (right), + (white) and - (black)

12/17 Stochastic mortality: Wills-Sherris model c ln(µ(x, t)) = ln(µ(x, t)) ln(µ(x 1, t 1)) = ax + b + σε(x, t) (8) µ(x, t): force of mortality for a cohort aged x at time t a, b and σ: parameters to be estimated ε(x, t): a standard normal variable Age dependence: [ε(x 1, t), ε(x 2, t),, ε(x N, t)] = Ω 1 2 Wt

Stochastic mortality: Wills-Sherris model (cont d) Parameter estimation: Parameter Male Female â ( 10 4 ) -2.36 4.02 ˆb ( 10 2 ) 8.38 4.59 ˆσ ( 10 2 ) 5.76 6.52 Age 100 95 90 85 80 75 70 65 60 55 1975 1980 1985 1990 1995 2000 2005 Time Age 100 95 90 85 80 75 70 65 60 55 1975 1980 1985 1990 1995 2000 2005 Time Males (left) and females (right), + (white) and - (black)

14/17 Projected survival curve: three models 1 0.8 Survival probability 0.6 0.4 0.2 0 Deterministic CBD, mean CBD, 90% CI WS, mean WS, 90% CI 70 80 90 100 110 Age Figure: Simulated survival probabilities of 65-year-old females.

15/17 Topic coverage 1 Introduction 2 Pricing Framework 3 Idiosyncratic House Price Risk 4 Termination Model 5 Pricing and Risk Analysis 6 Conclusions

NNEG and mortgage insurance premium (Age 65, LTV 0.4): house price risk

NNEG and mortgage insurance premium (Age 65, LTV 0.4): longevity risk

17/17 Topic coverage 1 Introduction 2 Pricing Framework 3 Idiosyncratic House Price Risk 4 Termination Model 5 Pricing and Risk Analysis 6 Conclusions

17/17 Conclusions Pricing and risk analysis of reverse mortgages idiosyncratic house price risk longevity risk Considerable house price risk using an aggregate index underestimates the risk risk factors associated with house characteristics should be used Longevity risk mortality improvement modelling has a large impact effect smaller than that of house price risk Improved insights into product pricing and risk analysis

References Alai, D., Chen, H., Cho, D., Hanewald, K., and Sherris, M. (2013). Developing equity release markets: Risk analysis for reverse mortgages and home reversions. UNSW Australian School of Business Research Paper No. 2013ACTL01. Chen, H., Cox, S. H., and Wang, S. S. (2010). Is the home equity conversion mortgage in the United States sustainable? Evidence from pricing mortgage insurance permiums and non-recourse provisions using the conditional Esscher transform. Insurance: Mathematics and Economics, 46(2):371 384. Cho, D., Sherris, M., and Hanewald, K. (2013). Risk management and payout design of reverse mortgages. UNSW Australian School of Business Research Paper No. 2013ACTL07. Cochrane, J. and Piazzesi, M. (2002). Bond risk premia. National Bureau of Economic Research (NBER) Working Paper, (9178). Hanewald, K. and Sherris, M. (2013). Postcode-level house price models for applications in banking and insurance. Economic Record, Forthcoming. Ji, M., Hardy, M., and Li, J. S.-H. (2012). A Semi-Markov Multiple State Model for Reverse Mortgage Terminations Johnny Siu-Hang Li. Annals of Actuarial Science, (Forthcoming). Lee, Y.-T., Wang, C.-W., and Huang, H.-C. (2012). On the valuation of reverse mortgages with regular tenure payments. Insurance: Mathematics and Economics, 51(2):430 441. Li, J. S. H., Hardy, M. R., and Tan, K. S. (2010). On Pricing and Hedging the No-Negative-Equity Guarantee in Equity Release Mechanisms. Journal of Risk and Insurance, 77(2):499 522. Shao, A. W., Sherris, M., and Hanewald, K. (2013). Disaggregated house price indices. UNSW Australian School of Business Research Paper No. 2013ACTL09. Wang, L., Valdez, E. A., and Piggott, J. (2008). Securitization of Longevity Risk in Reverse Mortgages. North American Actuarial Journal, 12(4):345 371. Yang, S. S. (2011). Securitisation and tranching longevity and house price risk for reverse

Appendix A: Australian market for reverse mortgages Figure: Australian Market for Reverse Mortgages, data source: Deloitte and SEQUAL media release 2012

Appendix B: NNEG and mortgage insurance premium Model Deterministic Wills-Sherris Cairns-Blake-Dowd LTV 0.2 0.4 0.6 0.2 0.4 0.6 0.2 0.4 0.6 A. Overall Sydney house price index π (p.a.) 0.003% 0.230% 3.246% 0.009% 0.360% 2.583% 0.003% 0.237% 3.126% NNEG 71 12,794 400,017 279 22,393 335,952 90 13,147 379,366 S.E. 17 498 2,131 36 639 2,038 18 491 2,094 TVaR 0.000 0.000 0.000 0.467 6.048 12.913 0.179 5.278 13.487 B. Houses near the central business district π (p.a.) 0.218% 0.720% 1.829% 0.239% 0.711% 1.621% 0.218% 0.716% 1.819% NNEG 6,043 42,421 186,092 7,298 46,370 181,302 6,036 42,138 184,776 S.E. 470 1,673 4,092 494 1,680 3,876 463 1,651 4,048 TVaR 0.000 0.000 0.000 6.654 17.148 29.594 6.424 17.779 31.168 C. Houses near to coastlines π (p.a.) 0.076% 0.255% 1.184% 0.088% 0.302% 1.183% 0.076% 0.257% 1.173% NNEG 2,062 14,238 110,932 2,624 18,645 124,031 2,070 14,284 109,598 S.E. 289 879 2,399 308 939 2,402 286 866 2,359 TVaR 0.000 0.000 0.000 4.387 11.923 21.331 3.893 11.512 22.120 D. Houses with less than or equal to two bathrooms π (p.a.) 0.010% 0.247% 3.078% 0.019% 0.374% 2.485% 0.011% 0.253% 2.968% NNEG 269 13,752 370,431 561 23,275 318,003 294 14,080 352,317 S.E. 86 566 2,239 99 692 2,148 87 558 2,198 TVaR 0.000 0.000 0.000 1.028 6.913 13.748 0.634 6.170 14.350 E. Houses with more than two bathrooms π (p.a.) 0.058% 0.418% 2.868% 0.081% 0.540% 2.376% 0.059% 0.420% 2.781% NNEG 1,577 23,759 335,272 2,412 34,438 298,788 1,612 23,871 321,653 S.E. 209 893 2,871 232 1,005 2,717 205 874 2,807 TVaR 0.000 0.000 0.000 3.391 10.145 17.505 2.914 9.912 18.420